Question: The integer $x$ has 12 positive factors. The numbers 12 and 15 are factors of $x$. What is $x$?
Since $12=2^2\cdot3$ and $15=3\cdot5$ are factors of $x$, $x$ must be divisible by the least common multiple of 12 and 15, which is $2^2\cdot3\cdot5$. Since $x$ has 12 factors and the LCM has $(2+1)(1+1)(1+1)=12$ factors, $x=2^2\cdot3\cdot5=\boxed{60}$.